ATTENTION:


This is my old webpage. As of June 27, 2005, this webpage will no longer be kept up to date and it will most likely disappear completely in December 2005. My current webpage is www.math.lsa.umich.edu/~speyer.
David E Speyer's Home Page

David E Speyer

You can contact me by:


E-mail: speyer@post.harvard.edu or speyer@umich.edu
The former is a forwarding address that currently forwards to the latter. The former will always be correct, but has problems with large files. The latter is my current e-mail address at the University of Michigan. I am currently in the process of phasing out my old Berkeley e-mail account, which is why it is no longer listed here. However, if you have sent something to that address, do not worry -- it will be forwarding to the Michigan address until December 2005 and I will also check it directly at least once a week.

Phone (Cell): (734)-255-8610

Mail (Home, After Mid-July):
David E Speyer
332 John Street
Ann Arbor, MI
48104-2919 USA
Map

Mail (Before Mid-July):
I will be moving to the above listed address in mid-July. Until then, the best way to reach me by mail will be to write to
David E Speyer
c/o Erin Larkspur
2195 Steeplechase Drive
Ann Arbor, MI 48103-6032 USA
And if you know Erin, write to her too!

Shouting:
My last name is pronounced "spire", like the top of a church. But I'll answer to "Sp-vowel-r!" or to "David!". For the curious, it is derived from the city of Speyer, Germany, although my friend Aaron tells me it could also be Yiddish for "pistol".

About Me:

I am a visiting scholar in the department of mathematics at the Univeristy of Michigan. (This webpage will be migrating there soon). I am funded by a Five Year Clay Research Fellowship.

I recently graduated from UC Berkeley, where I was a student of Bernd Sturmfels. My thesis is on tropical geometry, an approach to turning algebraic geometry problems into polyhedral geometry. Before coming to Berkeley, I was an undergraduate at Harvard. While there, I worked for Jim Propp's research group REACH and wrote an undergraduate thesis on the Eichler-Shimura correspondence under William Stein. I spent the rest of my time doing technical theater, hanging out with science fiction fans and working on Les Phys -- the physics musical! I spent four years as a counselor at PROMYS, a number theory program for high school students, and highly endorse it either as a place to go learn or to go work. I spent my own high school summers at MOP , which I found great but works better for some people than for others. I went to High School at Choate Rosemary Hall and to Middle School at Talcott Mountain Academy. If you are a young nerd in Connecticut, looking for a middle school or after school program, I highly recommend Talcott.

I enjoy algebraic problems with a combinatorial flavor. If you started out with a well motivated algebraic question but wound up with lots of little complicated pictures, I probably want to hear about it. Some topics I can usually be relied upon to think about: tropical geometry, cluster algebras, flag manifolds and other geometry of Lie Groups, interesting degenerations of algebraic structures, exact results and asymptopics of perfect matchings ( eg Arctic Circle phenomena). I also know a reasonable amount of Number Theory and enjoy talking about it, although as yet this is not a research interest.

If you want to know more about me, you can read my cv, see what I told the NSF I would work on or what I would have told them had there been no page limit. The second version is has some modifications reflecting later developments on these projects. Alternatively, you can see what you can figure out from my friends' websites.

In other news, my team Physical Plant has finally won the MIT Mystery Hunt. That means we get to write next year's hunt. Mwa ha ha...

Papers:

All of my papers that are in a reasonably polished state can be found here on the arXiv . (With the exception of my first paper, "Every tree is 3-equitable" with Zsuzsanna Szansiszlo.)
Here is a complete annotated listing of my papers as of June 7, 2005.

  • Tropical Geometry
    This is my dissertation, which attempts to do the ground work to establish tropicalization as a major tool of algebraic geometry. There are four major sections (plus a historical introduction.) The first section tries to develop general tools, including establishing the equivalence of several notions of tropicalization and describing the tropical degeneration and compactification -- these are schemes assosciated to a subvariety of a torus over a nonarchimedean field. The combinatorics of these schemes are indexed by a polyhedral complex whose underlying point set is the tropicalization. I have done more work here then is in the written version; I hope to eventually publish a paper on this subject, possibly in collaberation with Paul Hacking. The second section and third section respectively cover the material in my papers "The Tropical Grassmannian" and "Tropical Linear Spaces" below, rewritten to emphasize their connections to the other material of the dissertation. The final section studies the probleming of recognizing which graphs embedded in R^n occur as tropicalizations of curves embedded in the torus. It turns out that Mumford's techniques of nonarchimedean uniformization are admirably suited to this problem. The curve material, with a few technical hypotheses removed, will probably appear in a seperate publication.
  • Tropical Linear Spaces
    I define tropical analogues of the notion of "linear space" and "Plucker coordinates" and basic constructions for working with them. This paper is an exhaustive introduction that tells almost everything I have figured out. The most interesting aspect of the paper is the f-vector conjecture -- I conjecture what the maximal possible f-vector of a tropical linear space should be and provide a great deal of evidence for this claim. Out of all the conjectures I have made, this is the one that frustrates me the most; I would really like to get it before I graduate.
    Although it can be read independently, this paper is naturally a sequel to my paper "The Tropical Grassmannian" below.
  • A Broken Circuit Ring with Nick Proudfoot
    Given a linear subspace of affine space, we study the ring of rational functions on the linear space generated by the reciprocals of the coordinate functions. This ring has been studied previously by Terao and others. We find a universal Groebner basis and show that the ring degenerates to the Stanley-Reisner ring of the broken circuit complex.
  • Tropical Mathematics with Bernd Sturmfels
    An elementary introduction to Tropical mathematics, expanding on my co-author's Clay Public Lecture at Park City Math Institute 2004 (IAS/PCMI)
  • An arctic circle theorem for groves with Kyle Petersen
    Presented at Formal Power Series and Algebraic Combinatorics 2004.
    Journal of Combinatorial Theory: Series A 111 Issue 1 (2005), p. 137-164
    Proves that a randomly chosen grove (introduced in my paper with Gabriel Carroll below) is "frozen" outside a certain circle. This is analogous to results on random tilings of Aztec Diamonds and random Alternating Sign Matrices.
  • The tropical totally positive Grassmannian with Lauren Williams
    Accepted, Journal of Algebraic Combinatorics
    We study the tropical analogue of the totally positive cell in the Grassmannian, introduced by Lusztig and studied in detail by Postnikov and others. We discover a tight connection to the combinatorics of cluster algebras and conjecture a general connection between the cluster complex of a cluster algebra and its totally positive tropicalization.
  • Horn's Problem, Vinnikov Curves and Hives
    Duke Journal of Mathematics 127 no. 3 (2005), p. 395-428
    Horn's Problem asks to characterize the possible eigenvalues of a triples of Hermitian matrices with sum 0. Allen Knutson and Terry Tao gave an answer in terms of combinatorial objects called honeycombs which look like tropical curves. I explain this phenomenon by showing that Horn's problem is equivalent to studying the possible intersections of plane curves with prescribed topology with the coordinate axes and then showing that the tropical version of this criterion recovers the results of Knutson and Tao.
  • Reconstructing Trees from Subtree Weights with Lior Pachter
    Applied Mathematical Letters, 17 (2004), p. 615-621
    In computational phylogenetics, the problem of reconstructing a metric tree from the distances between its leaves frequently arises. We study the similar problem of reconstructing a tree from the total lengths of the subtrees spanned by k of its leaves.
  • The Tropical Grassmannian with Bernd Sturmfels
    Advances in Geometry, 4 (2004), no. 3, p. 389-411
    We study the tropicalization of the Grassmannian in its standard Plucker emebedding. We show that its points parameterize tropicalizations of linear spaces, give a complete description of the case of G(2,n) and do some computations of larger cases.
    I have done a good deal more work on the properties and classification of tropicalizations of linear spaces, see my paper "Tropical Linear Spaces" above.
  • Perfect Matchings and the Octahedron Recurrence
    The octahedron recurrence is a certain recurrence whose entries are indexed by a three dimensional lattice; the recurrence grows from a two dimensional surface of initial conditions. It follows from Fomin and Zelevinski's results on Cluster Algebras that all of the terms of the recurrence are Laurent polynomials in the initial values. I show that every term in these polynomials has coefficient 1 by establishing a bijection between these monomials and the perfect matchings of certain graphs. Special cases include formulas for Somos-4 and Somos-5 and for the number of perfect matchings of many families of graphs.
  • The Cube Recurrence with Gabriel Carroll
    Elec. Jour. of Comb., 11 (2004) #R73
    This paper is similar to the octahedron recurrence paper, but with applications to Propp's cube recurrence, a peculiar recurrence that has Laurentness and positivity properties similar to the octahedron recurrence but has no known relation to cluster algebras. The relevant combinatorial objects are no longer perfect matchings but "groves", certain highly symmetric forests that deserve further study.
  • "Every tree is 3-equitable" with Zsuzsanna Szansiszlo
    Discrete Math.,220, (2000) 283-289
    Let G be a graph whose vertices are labelled with the numbers 0, 1, ... i. Label each edge with the absolute value of the difference between its endpoints. A labelling is called equitable if, for any two numbers a and b from 0 to i, the number of vertices with label a differs by at most one from the number with label b and a similar property holds for the number of edges with each label. It is conjectured that every tree has an equitable labelling for every i. We prove this conjecture for i=2.